From the introduction, a Number Pyramid is composed of the 10 numbers
0-9. By clue 1, the sum of the two numbers in the second row equals 11, so
that the leftmost number must be 2 or greater, since the rightmost number
cannot be 10 or 11. By clue 4, the middle number in the third row minus the
leftmost number in the second row equals 4. The leftmost number in the
second row therefore must be 5 or less, since the middle value in the third
row can be no more than 9. So, the leftmost number in the second row is 2,
3, 4, or 5. If the leftmost number in the second row were 2, the other number
in the second row would be 9 (clue 1), the middle number in the third row
would be 6 (clue 4), and the top number would be 5 (5). By clue 2, the sum of
the numbers in the bottom row minus the sum of the numbers in the third row
equals 10. Since the numbers left are 0, 1, 3, 4, 7, and 8, none of 4, 7, and
8 could be in the third row with 6 because no combination of remaining numbers
would add to a sum 10 greater than the second row sum. If the 3 were in the
second row with 6, then 4, 7, and 8 would sum to 19--but the odd 1 couldn't go in
either row, so 3 couldn't be in the second row either. Adding the 0 and 1 to
6 would give 7, but the fourth row would add to 22, contradicting clue 2.
Therefore, the 2 cannot work as the leftmost value of the second row. If 4
were the leftmost number in the second row, 7 would be the rightmost (1),
8 would be the number in the middle of the third row (4), and 3 would be the
number on top the Number Pyramid (5). Again by clue 2, the sum of
the numbers in the bottom row minus the sum of the numbers in the third row
equals 10, using 0, 1, 2, 5, 6, and 9. If any of 5, 6, or 9 were in the
second row with 8, there would be no way for clue 2 to work; so those numbers
would be in the bottom row, giving at least 20. Then the 2 would have to go
in the second row with 8 to get 10--but the odd 1 cannot fit. So, the leftmost
number in the second row isn't 4. If the leftmost number in the second row
were 5, 6 would be the rightmost (1), 9 would be the number in the middle of
the third row (4), and 2 would be the number on top the Number Pyramid
(5), leaving 0, 1, 3, 4, 7, and 8. Again by clue 2, the sum of the numbers in
the bottom row minus the sum of the numbers in the third row equals 10. None
of 3, 4, 7, or 8 could be with 9 in the third row because no combination of
remaining numbers in the fourth row could satisfy clue 2. With the 0 and 1
being in the second row, however, the sum of 10 would be 12 short of the 3, 4,
7, and 8 in the bottom row, contradicting clue 2. So, the leftmost number in
the second row isn't 5 and must be 3. Then 8 would be the rightmost (1), 7
would be the number in the middle of the third row (4), and 4 would be the
number on top of Number Pyramid 2. Again by clue 2, the sum of the
numbers in the bottom row minus the sum of the numbers in the third row equals
10. Of the remaining numbers, neither 5, 6, or 9 could be in the third row
or clue 2 couldn't work, so those three numbers add to 20 in the bottom row.
In order for clue 2 to be satisfied, the 1 and 2 must go in the third row to
sum to 10 and the 0 must go in the bottom row to keep that sum at 20. By
clue 3, the righthand four numbers sum to 18, with 4 in the top and 8 in the
second row leaving 6 for the bottom two rows. The only combination that works
is 1 as the rightmost number in the third row and 5 as the rightmost number in
the fourth row. By elimination, then, 2 is the leftmost figure in the third
row. By clue 6, the 0 must be the number next to the 5; while the 9 must be
the leftmost number and the 6 next to the 9. In sum, Number Pyramid 2
is as follows: